A system of uniform rod of mass m and length 2R and a uniform disc of m and radius R are as shown in the figure. The moment of inertia of the system about the axis `OO^' ` (perpendicular to plane of disc) will be

Options

A. `(79mR^2) /12`
B. `(41mR^2) /6`
C. `(7mR^2) /6`
D. `(25mR^2) /12`

Answer

To find the moment of inertia of the system consisting of a uniform rod and a uniform disc about the axis OO' (perpendicular to the plane of the disc), we will calculate the moment of inertia of both components separately and then sum them up.

### Step-by-Step Solution:

1. **Identify the Components**:
- We have a uniform rod of mass \( m \) and length \( 2R \).
- We have a uniform disc of mass \( m \) and radius \( R \).

2. **Moment of Inertia of the Rod**:
- The moment of inertia \( I \) of a rod about an axis perpendicular to its length and passing through one end is given by:
\[ I_{\text{rod}} = \frac{1}{3} m L^2 \]
- Here, \( L = 2R \), so substituting in:
\[ I_{\text{rod}} = \frac{1}{3} m (2R)^2 = \frac{1}{3} m (4R^2) = \frac{4}{3} m R^2 \]

3. **Moment of Inertia of the Disc**:
- The moment of inertia \( I \) of a disc about an axis through its center and perpendicular to its plane is given by:
\[ I_{\text{disc}} = \frac{1}{2} m R^2 \]
- However, since the axis OO' is not through the center of the disc but at a distance \( 2R \) from the center (the length of the rod), we need to use the parallel axis theorem:
\[ I' = I + md^2 \]
- Here, \( d = 2R \), so:
\[ I'_{\text{disc}} = \frac{1}{2} m R^2 + m (2R)^2 = \frac{1}{2} m R^2 + 4m R^2 = \frac{1}{2} m R^2 + 8m R^2 = \frac{17}{2} m R^2 \]

4. **Total Moment of Inertia**:
- Now, we can find the total moment of inertia \( I_{\text{net}} \) of the system about the axis OO':
\[ I_{\text{net}} = I_{\text{rod}} + I'_{\text{disc}} = \frac{4}{3} m R^2 + \frac{17}{2} m R^2 \]
- To add these fractions, we need a common denominator, which is 6:
\[ I_{\text{net}} = \frac{8}{6} m R^2 + \frac{51}{6} m R^2 = \frac{59}{6} m R^2 \]

5. **Final Result**:
- Therefore, the moment of inertia of the system about the axis OO' is:
\[ I_{\text{net}} = \frac{59}{6} m R^2 \]